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Computing zonoid trimmed regions of dimension d>2

Author

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  • Mosler, Karl
  • Lange, Tatjana
  • Bazovkin, Pavel

Abstract

A probability distribution on Euclidean d-space can be described by its zonoid regions. These regions form a nested family of convex sets around the expectation, each being closed and bounded. The zonoid regions of an empirical distribution introduce an ordering of the data that has many applications in multivariate statistical analysis, e.g.cluster analysis, tests for multivariate location and scale, and risk analysis. An exact algorithm is developed to constructing the zonoid regions of a d-variate empirical distribution by their facets when d>=3. The vertices of the region and their adjacency are characterized, and a procedure is suggested by which all vertices and facets can be determined. The algorithm is available as an R-package.

Suggested Citation

  • Mosler, Karl & Lange, Tatjana & Bazovkin, Pavel, 2009. "Computing zonoid trimmed regions of dimension d>2," Computational Statistics & Data Analysis, Elsevier, vol. 53(7), pages 2500-2510, May.
    • Handle: RePEc:eee:csdana:v:53:y:2009:i:7:p:2500-2510
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    References listed on IDEAS

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    1. Gleb A. Koshevoy & Karl Mosler, 2007. "Multivariate Lorenz dominance based on zonoids," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 91(1), pages 57-76, March.
    2. Ignacio Cascos & Ilya Molchanov, 2007. "Multivariate risks and depth-trimmed regions," Finance and Stochastics, Springer, vol. 11(3), pages 373-397, July.
    3. Ruts, Ida & Rousseeuw, Peter J., 1996. "Computing depth contours of bivariate point clouds," Computational Statistics & Data Analysis, Elsevier, vol. 23(1), pages 153-168, November.
    4. Lopez-Pintado, Sara & Romo, Juan, 2007. "Depth-based inference for functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4957-4968, June.
    5. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
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    Cited by:

    1. Wiechers, Christof, 2011. "Construction of uncertainty sets for portfolio selection problems," Discussion Papers in Econometrics and Statistics 4/11, University of Cologne, Institute of Econometrics and Statistics.
    2. Xiaohui Liu & Shihua Luo & Yijun Zuo, 2020. "Some results on the computing of Tukey’s halfspace median," Statistical Papers, Springer, vol. 61(1), pages 303-316, February.
    3. Liu, Xiaohui & Zuo, Yijun, 2015. "CompPD: A MATLAB Package for Computing Projection Depth," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 65(i02).
    4. Xiaohui Liu, 2017. "Fast implementation of the Tukey depth," Computational Statistics, Springer, vol. 32(4), pages 1395-1410, December.
    5. Zuo, Yijun, 2021. "Computation of projection regression depth and its induced median," Computational Statistics & Data Analysis, Elsevier, vol. 158(C).
    6. Bazovkin, Pavel & Mosler, Karl, 2012. "An Exact Algorithm for Weighted-Mean Trimmed Regions in Any Dimension," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 47(i13).
    7. Dyckerhoff, Rainer & Mozharovskyi, Pavlo, 2016. "Exact computation of the halfspace depth," Computational Statistics & Data Analysis, Elsevier, vol. 98(C), pages 19-30.
    8. Chiara GIGLIARANO & Karl MOSLER, 2009. "Measuring middle-class decline in one and many attributes," Working Papers 333, Universita' Politecnica delle Marche (I), Dipartimento di Scienze Economiche e Sociali.
    9. Karl Mosler, 2020. "Commentary on “From unidimensional to multidimensional inequality: a review”," METRON, Springer;Sapienza Università di Roma, vol. 78(1), pages 51-54, April.
    10. Liu, Xiaohui & Rahman, Jafer & Luo, Shihua, 2019. "Generalized and robustified empirical depths for multivariate data," Statistics & Probability Letters, Elsevier, vol. 146(C), pages 70-79.

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